Hyperspheres Admitting a Pointwise Symmetry Part 1

An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(TpM) for all p ∈ M , which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i. e. S = HId (and thus S is trivially preserved). First we solve an algebraic problem. We determine the non-trivial stabilizers G of a traceless cubic form on a Lorentz-Minkowski space R1 under the action of the isometry group SO(1, 2) and find a representative of each SO(1, 2)/G-orbit. Since the affine cubic form is defined by h and K, this gives us the possible symmetry groups G and for each G a canonical form of K. In this first part, we show that hyperspheres admitting a pointwise Z2×Z2 resp. R-symmetry are well-known, they have constant sectional curvature and Pick invariant J < 0 resp. J = 0. The classification of affine hyperspheres admitting a pointwise G-symmetry will be continued elsewhere. MSC2000: 53A15 (primary), 15A21, 53B30 (secondary)